Series of gear speed reducers

ABSTRACT

The invention relates to a series of n gear speed reducers with at least n unequal center distances in increasing order, the absolute values of the last center distance of each speed reducer in the series, i.e. of the specific center distances, form an improper geometric progression of which the successive values of the ratio form a geometric progression.

I Umted States Patent [151 3,673,885 Hansen July 4, 1972 1 SERIES OFGEAR SPEED REDUCERS References Cited [72] Inventor David Hansen,Mortsel, Belgium UNITED STATES PATENTS 73 Assigneez Machinery and tHansen Edegem l,728,279 I 9/1929 Ramsey ..74/325 Be|gium 3,358,52512/1967 Clarke ..74/325 X [22] 1970 Primary ExaminerLeonard H. Gerin[21] Appl. No.: 94,439 Attorney-Bacon & Thomas [30] Foreign ApplicationPriority Data [57] ABSTRACT March 11, 1970 Belgium ..747171 Theinvention relates to a Series gear Speed reducers with at least nunequal center distances in increasing order, the ab- [52] US. Cl...74/421 R, 74/325 solute values of the las center distance of eachspeed reducer [51] Int. Cl r ..Fl6h 1/20, Fl6h 3/08 in the series, i.e.of the specific center distances, form an im- [58] Field of Search..74/4l2 R, 42] R, 325 proper geometric progression of which thesuccessive values of the ratio form a geometric progression.

18 Claims, 14 Drawing Figures PATENTEDJUUMQR 3.673.885

sum 10; s

INVENTOR.

04 W0 6? Hen/5m BY H TTOF/V Y5 PATENTEDJUL 4 m2 3.673 .885 SHEET 2 OF 5I N VE N TOR. 0H W0 6. HANSEN HTTVRNE Y5 PATENTEIJJUL 41972 sum 30; s

I N VE N TOR 00 W0 C. H/I/VSEN ATTORNEYS PATENTEIJJUL 4 m2 SHEET 5 W 5INVENTOR. 0.4 v10 6. HANSEN BY ATTORNEYS SERIES OF GEAR SPEED REDUCERSThis invention has as its object a series of gear speed reducersfulfilling the optimal conditions for rational and profitableprefabrication, meeting at the same time the interests and convenienceof the manufacturer and the interests and requirements of the user. Theseries in question is conceived in such a way that each speed reducer ina given series draws near to the specific requirements of the user in anideal way. These requirements principally that dominate the problem.

The benefits of a rational prefabrication for the manufacturer are infact well known. These benefits will be all the more saveguarded as willbe the interests and requirements of the user. For this purpose,starting from options judiciously taken by the manufacturer, the usermust be able to find in the given series, the appropriate reducer orreducers, which approximate most closely the ideal conditions, free fromany under-dimensioning and with a least amount of overdimensioning ofthe components within a reducer with respect to one another.

These options lead the manufacturer to (1) choose beforehand amanufacturing program, determined by the well-considered choice of thesmallest and the largest sizes; (2) find a judicious law of progressionbetween the two limits of the program established at (1); (3) decideupon the number of sizes, considered to be sufficient to meet thedemands of the market.

In presence of these options, the manufacturer finds himself facing theproblem of the execution of all and every one of the gear speed reducersin the chosen series, an execution which must at all times be adequateand economical. Each reducer consists mainly of a housing and of gearsets. The problem of execution consists of imagining a solution, withinthe above mentioned options, which enables an optimal use of the samehousings and the same gear sets in a maximum number of validcombinations, i.e. combinations which meet practical requirements, thereducers having in none of the many combinations components which areneedlessly overdimensioned with respect to one another.

In other words, the series has to be conceived in such a way that,although it consists of prefabricated material, each reducer closelyapproximates that, which would have been adopted if each case had beencalculated separately.

These requirements tie the prefabrication of both housing and gears tocertain conditions. The series of gear speed reducers, according to theinvention, is therefore characterized by very particular rules, on theone hand for the. establishment of a series of center distances and inthe framework of options of a manufacturing nature the spacing of thesizes of the gears suitable to the series considered; and, on the otherhand for the judicious choice, in this spacing, of the center distancesand thus of the gears for each size of reducer in the given serres.

For more clarity, reference will always be made hereafter, and only asan example, to a series of reducers in which each reducer has threeunequal center distances of increasing size. The last center distance ineach reducer of the series, is called hereafter specific centerdistance."

The specific center distance of a speed reducer can be the first orsecond center distance of other reducers in the series, so that thenumber of center distance'swith different absolute values, may be veryclose to the number of reducers in the given series, without being less.

Therefore, according to the invention, the series of gear speed reducersis essentially characterized by the fact that the absolute values of thelast center distance of each reducer in the series or specific centerdistance constitute an improper geometric progression, the successivevalues of whose ratio constitute a proper geometric progression.

On the other hand, again according to the invention, the series of gearspeed reducers in question is characterized by the fact that eachreducer of the series comprises gear sets such that the values of allprima'ry and overall reduction ratios of said reducers are part of thesame geometric progression.

For more clarity, use will be made hereunder of a model program ofprefabrication of a series of gear speed reducers, according to theinvention, on the understanding that it only has to be considered as anexample, without restricting the object and the scope of the invention.

In the enclosed drawings,

FIGS. 1 to 10 show, schematically and as a front view, the successivereducers of the series considered;

FIG. 11 represents a table giving the selection of the specific centerdistances in the execution considered;

FIG. 12 represents schematically and in perspective, a gear speedreducer with parallel input and output shafts;

FIG. 13 represents schematically and in perspective, a gear speedreducer with perpendicular input and output shafts;

FIG. 14 represents schemafically and in perspective, a gear speedreducer with a built-on motor, whose axis constitutes the primary axisof the reducer.

In the example chosen, the series comprises reducers only with helicalgears and three unequal center distances of increasing size, the thirdcenter distance being called the specific center distance. The number nof reducers A, B, C, D, E, F, G, H, K, L (FIGS. 1 to 10) in the seriesequals 10, whereas the number of different center distances of theseries in question is n 2 12, designated by the series a a a,, a a a Thenumber of specific center distances also equals 10 and each of thespecific center distances i.e. a to a,,,, may be found as a first, asecond or a third center distance in the reducers of the series. Asshown in the table of FIG. 11, the center distances a to 0 are usedrespectively 1, 3, 3, 4, 5, 3, 4, 2, 2, l, I, and l times, which meansthat there are some 30 center distances obtained from 1 2 differentcenter distances.

If this repetition in the use of the center distances is advantageous,still more is the way in which the range of associated torques, isestablished. This determines the range of characteristic torques of thespeed reducers in the series in question, so as to meet the requirementsof the user better, and also the way in which these center distances areselected in such a way as to harmonize with the center distances, usedas specific center distances, in order to obtain as favorable aspossible a power equilibrium.

To make this more understandable, the absolute values of the tenspecific center distances a up to a in the series of the ten gear speedreducers may be considered to form a geometric progression in which thecommon ratio h is itself a variable, i.e. an improper geometricprogression, and the successive values of h constitute a propergeometric progression, whose constant ratio is equal to p. The valuesa,h and p are calculated in function of the characteristic torques ofthe different sizes, which lies in the sphere of any qualified personfamiliar with the manufacture and the sale of gear speed reducers.

The series of center distances a to a is represented as follows, takinginto consideration that a, is any given center distance of the series inquestion:

As to the series of variable common ratios h, it is represented asfollows, taking into consideration that h, is any given variable commonratio in the series:

r sfpi n-1 n-2 p which can also be written:

Considering series (I) the values of h in function of the centerdistances a may be deduced from it:

From (3) the following relationship is deduced:

Combining the two series l and (2), one deduces that the value of anycenter distance a, can be calculated from the relationship.

This shows that any qualified person can easily apply the essentialcharacteristic of the invention, i.e. take care that, in the series, theabsolute values of the specific center distances a a a,, form animproper geometric progression of which the variable common ratio itselfh h h,, follows the law of a proper geometric progression with constantcommon ratio p.

Between the face width b of the gear and the center distance a, thefollowing relationship can be established:

b K a (8 which is valid for all the gears in the series in question andin which, for a given reduction ratio, K and e are constants, which canbe determined by experience in the manufacture of gears, in such a waythat these gears can be produced with the necessary precision.

On the other hand and in order to approach the optimal conditions statedin this document, the characteristic torques of the successive sizes arechosen in such a way that they equal the characteristic torques of thelast center distance of the reducer, i.e. of the specific centerdistance. Moreover, the utmost has been done to bring close together thetorques, calculated in function of surface durability and of strength,establishing however the fact that the characteristic torque isdetermined by the calculation of the surface durability.

In these conditions, the characteristic torque T calculated according toa given process of calculation (in the example, the process put forwardby A.G.M.A. American Gear Manu facturers Association) may be expressedby the relationship;

T= K a b" 9 in which If is a constant exponent for all characteristictorques, being function of the raw material, the reduction ratio and therotational speed of the secondary shaft; a is the center distance, bindicates the face width of the gears, f and g are constant numbers.

The characteristic torque T, of any given specific center distance a,may be written, according to the relationships (8) and (9):

in which K and m are constants with K K K and m f e g In this way aseries of characteristic torques may be placed alongside the series ofcenter distances, so that the terms of both series correspond two bytwo. From the relationship 10) between T, and a,, it is established thatthe series of characteristic torques is, as well as the series ofspecific center distances, a geometric progression having identicalproperties.

The series of characteristic torques Tmay be written as follows; takingin consideration that 7} is any given characteristic torque taken fromthe series T up to T,,:

By comparison with the series of specific center distances (l) andtaking the relationship (10) into account, one can write:

l-l l (12) The series of the variable ratios k is, as is also series(2), a geometric progression.

Indeed:

from (12) results k =h from (3) results h;=h p*" therefore k =h (p from(12), is found h =k or also h =VE Taking g=p in which q is constant or,also p=W one can finally write:

r iq which shows that the series of ratio k is a geometric progression.

The series of variable ratios k is expressed as follows, taking intoconsideration that k, represents any given ratio from the Taking therelationship (16) into consideration, one can Combining the series ofthe torques T (11) and of the ratio lo (16), one obtains:

T T k1 q 2 When i=n, the relationship (18) becomes:

T T k q 2 from which (nl)z(n-V qT m (20 Replacing g by the relationship(17) n E (nl)(n2) T n- 2 n 1 N 1 1 One finds that, taking the abovementioned options as a starting point, the numerical values of thespecific center distances as well as of the characteristic torques, maybe determined from the above stated relationships.

It will be observed too, that the choice of the options is important.

It is a function of the manufacturing program. As an example it will beobserved that the applicant has, among other things, drawn up a completeprogram having as its object the integral prefabrication of a series often sizes of reducers of medium capacity, of which the characteristictorque of the smallest size is equal to mkg (7,800 lbs.in) and thecharacteristic torque of the largest size is equal to 7,000 mkg (680,000lbs.in).

In the series in question, as already stated, one has, from the start,established that T 90 mkg (7,800 lbs.in) and T 7,000 mkg (680,000lbs.in).

Based on the data of experience, the relationship T /T k 2 between thesmallest torque T and the second torque T has been established.

On the other hand, taking into account the best conditions of adequacy,one has admitted that the characteristic torque T, of the size n isapproximatively equal to 1,3 times the characteristic torque of sizen-l, which gives the following relationship:

The ratio k,, which is the relationship between two successivecharacteristic torques, thus varies by application of what precedes in acontinuous and known way from one extwo center distances, selected fromthe series of specific center distances, shown above.

When considering a speed reducer, the permissible capacities of theincorporated gear sets can be compared to one Heme value to the otherone another and the most economical speed reducer will naturally beobtained, when the various characteristic capacities are as By choosingvalues 2 and m the Senes Ofk close as possible to one another and aimtowards equality. lnz i 29 21 ht h T T k d deed, in this case, oneavoids as much as possible, that one mm re a ons m w c 1 an are gear setshould be overdimensioned in regard of the other one. knoym from the0pm) taken value 9 be deduced 10 In terms of torque values, it may besaid, that the most as a general rule find a fracnonal value foreconomical speed reducer will be obtained, when the value of one W1choose, for the number of sizes, the nearest full the torque of thepenultimate center distance is as close as number. By then making thiscalculation in reverse, the exact possible to but not less than thevalue the chant} value of k,, will be established, and this will beadopted in lieu tetistic torque of the stat of the specific centerdistance in of the approx'mate ,Value ongmany fixed" the given speedreducer, divided by the reduction ratio used in lndeed, one obtarns forn 10.1 1 that center distance From this, one deduces that 10 sizes mustbe provided and o will proceed the same way f the other center onecalculatfis the exact value of 1.3 it?- distances. The reduction ratioof the last center distance plays it, according to the example, thefollowing values are ada part in establishing the criterion ofequilibrium or ofintemal mmed: economy of the speed reducer, variousratios being possible.

On the other hand, by repeated use'of the gears, the gears I f: used incenter distances, other than the specific center 8 distance, will bechosen from the series of specific center the gs (8) (9) and 10) becomerespecnvely: distances. Since, in order to create an adequate range ofspeed i 09 reducers, this series must follow a special law (an improperi g geometric progression), it is understandable that the technique ofthis choice must take into account also the law On determining thevalues of the constants K and K the mentioned with a certain amountofflexibihty values of h,, k,, p and q may be calculated from therelationone cans reduced torque" the torque at the pinion of the Ships(16) (13) and (20) respecnvely' whereas the senes specific centerdistance, of which the characteristic torque is of center distances aand the series of characteristic torques T of course the torque at thewheel, one can write: are easily determined by means of therelationships (7) and i 1 (18) respectively. 7 r, T/ R'- 01 19 1 Thevalues of the characteristic torques T of the successive T,. is theabove described reduced torque sizes thus form a geometric progression,whose ratio k itself is Tthe characteristic torque of the last centerdistance variable. R, any chosen reduction ratio of the last centerdistance.

TABLE I In k2 k3 k4 k5 ks k1 kg kg TABLE 11 T1 T, T3 T4 T5 s T1 a o TieTABLE III in at aa s4 a1 a1 a1 a; a an Said improper geometricprogression has, as a variable ratio, nine terms of which the first andthe last are established beforehand. As already stated in connectionwith the characteristics of the center distances, the different valuesof the ratio of said improper geometric progression vary themselvesaccording to a proper geometric progression whose constant ratio isexpressed by the relationship:

Since this ratio may vary from R to R, the following is obtained:

r mar r min and r min r ma:

The choice of a center distance from the series of specific centerdistances must be done in such a way that its torque T is as close aspossible to but not less than the reduced torque T,

Thus T r man- It must be noted that the absolute value of the reductionratio R, plays a part in establishing the criterion of selection andthat the relationship R,- /R, is a measure of the disequilibrium in thecapacities introduced in the speed reducer.

The fact of keeping the relationship R, ,/R, to a minimum, thereforeaims directly at maintaining the interior harmony or the interioreconomy of the reducers by avoiding excessive deviations from thestarting-situation in question,

that situation being the result of a judicious equilibration of theratio R, and the available specific center distances.

The table of FIG. 11 illustrates the application of what precedes.

Although not essential, it is very desirable and required by an evolutedpractice anyhow, that the same range (or the only range) of reductionratios are found in all sizes of gear reducers. On the basis of thesevery considerations, it is also good practice to adopt, as numbers,values that are generally used, in other words, nonnalized numbers orstandard numbers.

In the present case the so-called series R is applied (seeRecommendation No. 497 from the lntemational StandardizationOrganization).

lt will be noted that it is sufficient for the primary reduction ratios(i.e. those realized in a given center distance) to be part of thestated series R 20, so that also the overall reduction ratios (obtainedby combining the primary ratios) are part of the same series.

Combining primary ratios in order to obtain overall ratios is howeverconditioned in the present case by the repeated use of the same gears inthe series of speed reducers (a rationalization which diminishes thenumber of different gears, whence a total economy). This repeated use isregulated by the law of selectivity of the specific center distances,which is itself the result of a search for an equilibrium of capacitiesin every reducer (which aims at obtaining an internal or intrinsiceconomy).

In addition to the above conditions, the hoped-for result, (i.e. a rangeof ratios which is complete and is the same in all sizes) must beobtained by a judicious choice, for each center distance, of primaryreduction ratios from the series of standard numbers adopted. Thus, itappears that, faced with this problem and having a series of standardnumbers at ones disposal, the most economical solution must be lookedfor.

This solution is based upon the fact that the differences between theorder numbers of the reduction ratios for the same center distance mustbe different from those found in the adjoining center distance.

lt will be remembered that the R 20 standard numbers form a geometricprogression in which the ratio s is The terms of this progression whichare taken into account, are obtained by taking a full and non negativepower of the ratio s.

If more particularly one writes that the respective terms are obtainedby s" (n being the exponent and called order number"), to each ordernumber corresponds naturally a term in the series R 20 and consequentlya reduction ratio.

The table on page 19 gives the relation: order number reduction ratio,for the ratios from 1 to 100.

The concern for economy, which is at the base of the presentconsiderations, can also be expressed by the notion of efficiency,understanding from this, that, for a given number of primary reductionratios, one must obtain, by combination, a number of different values,that is as close as possible to the ideal number. ln a speed reducerwith two center distances, that number is obtained by multiplying thenumbers of reduction ratios of both center distances mutually.

If one takes the concrete case of the specific and the penultimatecenter distances of the series of speed reducers, object of the presentinvention, it will be remembered that, on striving for an equilibrium ofthe capacities in each of those reducers, and although each centerdistance has several (generally more than two) predetermined reductionratios, only two ratios are used if this center distance is used as thespecific center distance. This is important for the establishment of thepractical rule for the optimal choice of the reduction ratios.

Finally, it must not be forgotten that one of the aims in the field ofthe reduction ratios is to avoid gaps for shortcomings in the overallseries.

All preceding considerations lead to the formulation of a practicalrule, according to which the choice of the reduction ratios of eachcenter distance in the series of center distances considered, will bemade by taking into account the selection of these center distances insuch a way that the differences between the order numbers of successivereduction ratios, classified in increasing order, shown as Series I 3,l, 3, l, 3... )andSeriesIl(...2,2,2,2,2,... )belongtothe specific centerdistance and the penultimate center distance of a speed reducerrespectively or conversely.

On respecting this rule, the number of combinations of differentreduction ratios obtained, equals the ideal or maximum number, whichmeans thus for the whole range a minimum of different gear sets.

In the case of gear speed reducers, the standard numbers stated must berealized naturally as reduction ratios, i.e. as ratios of full numbersof teeth. This means that the above mentioned values will be obtainedwith a certain tolerance, which does not affect at all the fundamentalvalue of the established rule.

Therefore, thanks to this new conception in the establishment of thecharacteristics of all and of each gear speed reducer of a freelydetermined series in particular, one places at the disposal of industry,whatever kind of industry it may be, speed reducers, which, thanks to arelatively small number of different parts, enable not only a maximumnumber of combinations, but also such combinations as to meet almost allrequired applications with middle-sized speed reducers, with aharmoniously spread range of reduction ratios, being amply dimensioned,and showing no unnecessary overdimensioning of the components in areducer with respect to one another.

It has to be noted too, that the very particular specialcharacteristics, revealed by the present invention, do not introduce anydifficulties in the possibilities of diversification of a series ofspeed reducers. Indeed, the new considerations that determine theabsolute values of the center distances as well as the gears, do notinfluence the outward shape of the housing, i.e. of all speed reducersin the series.

Evidently, considering the experience already acquired in the field ofthe gear speed reducer, it is estimated that a maximum efficiency willbe obtained if the rules stated are applied to speed reducers with aprism shaped housing, which facilitates the manufacture and lends itselfadmirably to manifold possibilities of execution and also thepossibilities of mounting.

By belonging to a series of speed reducers, which is itself subject tothe laws of mutual dependence, the object of the invention isessentially tied to the notion itself of a series of gear speedreducers. However, since each speed reducer considered individuallybenefits, in intrinsic value, from the advantages of the series inquestion, the invention concerns equally each speed reducer in such aseries, considered individually.

What we claim is:

1. Series of n gear speed reducers with at least it unequal centerdistances in increasing order, with the characteristic that the absolutevalues of the last center distance of each speed reducer in the series,i.e. of the specific center distances, form an improper geometricprogression of which the successive values of the ratio form a geometricprogres- BIO".

2. Series of n gear speed reducers according to claim I, with thecharacteristic that the absolute values of the n specific centerdistances form an improper geometric progression, whose successivevalues h,, h h,, of the variable ratio fonn a proper geometricprogression whose constant ratio p is obtained by the relationship 11-2lT'l any one specific center distance a, being obtained by therelationship 2 i l l P in which K is a constant, a represents the centerdistance of the set considered and e is a constant exponent to bedetermined.

4. Series of reducers, according to claim 3, consisting of IO reducers,with the characteristic that the said exponent e is chosen as 0.9.

5. Series of reducers, according to claim 4, consisting of reducers,characterized by the fact that the characteristic torque of the smallestreducer in the series equals 90 mkg (7,800 lbs. in) and thecharacteristic torque of the largest reducer equals 7,000 mkg (680,000lbs.in).

6. Series of reducers, according to claim 5, with the characteristicthat the relationship between the characteristic torque of the secondreducer in the series and the characteristic torque of the firstreducer, equals 2.

7. Series of reducers, according to claim 6, with the characteristicthat, for a series of 10 reducers, the relationship between thecharacteristic torque of any one reducer and the characteristic torqueof the preceding reducer in the series, is obtained by one of the termsof a geometric progression, of which 2 is the first term and theconstant ratio is vided for the specific center distance of the reducerin.

question, and the minimum reduction ratio, used in the stated lastcenter distance.

9. Series of gear speed reducers, according to claim 1, with thecharacteristic that the values of all primary and overall reductionratios of the stated reducers are part of the same geometricprogression.

10. Series of gear speed reducers, according to claim 9, with thecharacteristic that the geometric progression stated, has a ratio ofs=vl0 the values in question being taken from the standard numbers 1 toof the progression in question.

11. Series of reducers, according to claim 9, with the characteristicthat the differences between the order numbers of the reduction ratios,belonging to the same center distance, are difierent from those whichcan be found in an adjoining center distance.

12. Series of reducers according to claim 11, with the characteristicthat the choice of the reduction ratios of each center distance is madein such a way that the differences between the order numbers of thesuccessive reduction ratios, classified in increasing order, shown asseries I 3, l, 3, l, 3 and series II 2,2,2,2, 2 belong to the specificcenter distance and the penultimate center distance of a speed reducerrespectively or conversely.

13. Series of reducers according to claim 12, with the characteristicthat the values of the reduction ratios of the reducers, having two tothree reduction steps, are essentially comprised between 8/ l and 100/l.

14. Series of reducers, according to claim 13, with the characteristicthat the secondary shaft of at least one of the reducers in the series,is hollow.

15. Series of reducers, according to claim 14, with the characteristicthat the primary and secondary hollow shafts of one or all reducers areparallel.

16. Series of reducers, according to claim 14, with the characteristicthat the primary and secondary hollow shafts of one or all reducers areperpendicular.

17. Series of reducers, according to claim 14, with the characteristicthat the primary shaft of one or all stated reducers with hollow shaft,is replaced by flange motor.

18. In its capacity as a new industrial product, each gear speed reducerbeing part of a series of reducers, according to claim 1.

1. Series of n gear speed reducers with at least n unequal centerdistances in increasing order, with the characteristic that the absolutevalues of the last center distance of each speed reducer in the series,i.e. of the specific center distances, form an improper geometricprogression of which the successive values of the ratio form a geometricprogression.
 2. Series of n gear speed reducers according to claim 1,with the characteristic that the absolute values of the n specificcenter distances form an improper geometric progression, whosesuccessive values h1, h2 . . . hn of the variable ratio form a propergeometric progression whose constant ratio p is obtained by therelationship any one specific center distance ai being obtained by therelationship relationships in which a1 and h1 are chosen values. 3.Series of gear speed reducers according to claim 1, with thecharacteristic that the face width b of any gear set is obtained by therelationship b K1 ae in which K1 is a constant, a represents the centerdistance of the set considered and e is a constant exponent to bedetermined.
 4. Series of reducers, according to claim 3, consisting of10 reducers, with the characteristic that the said exponent e is chosenas 0.9.
 5. Series of reducers, according to claim 4, consisting of 10reducers, characterized by the fact that the characteristic toRque ofthe smallest reducer in the series equals 90 mkg (7,800 lbs. in) and thecharacteristic torque of the largest reducer equals 7,000 mkg (680,000lbs.in).
 6. Series of reducers, according to claim 5, with thecharacteristic that the relationship between the characteristic torqueof the second reducer in the series and the characteristic torque of thefirst reducer, equals
 2. 7. Series of reducers, according to claim 6,with the characteristic that, for a series of 10 reducers, therelationship between the characteristic torque of any one reducer andthe characteristic torque of the preceding reducer in the series, isobtained by one of the terms of a geometric progression, of which 2 isthe first term and the constant ratio is T1 and Tn being respectivelythe first and the last characteristic torque.
 8. Series of gear speedreducers, according to claim 1, in which each reducer includes gearsets, provided for the center distances stated, with the characteristicthat in each reducer the value of the penultimate center distance isthat of the values still available, values whose characteristic torqueis as close as possible to, but not less than, the value given by therelation between the characteristic torque of the gear set, provided forthe specific center distance of the reducer in question, and the minimumreduction ratio, used in the stated last center distance.
 9. Series ofgear speed reducers, according to claim 1, with the characteristic thatthe values of all primary and overall reduction ratios of the statedreducers are part of the same geometric progression.
 10. Series of gearspeed reducers, according to claim 9, with the characteristic that thegeometric progression stated, has a ratio of the values in questionbeing taken from the standard numbers 1 to 100 of the progression inquestion.
 11. Series of reducers, according to claim 9, with thecharacteristic that the differences between the order numbers of thereduction ratios, belonging to the same center distance, are differentfrom those which can be found in an adjoining center distance. 12.Series of reducers according to claim 11, with the characteristic thatthe choice of the reduction ratios of each center distance is made insuch a way that the differences between the order numbers of thesuccessive reduction ratios, classified in increasing order, shown asseries I ( . . . 3, 1, 3, 1, 3 . . . ) and series II ( . . . 2,2,2,2,
 2. . . ) belong to the specific center distance and the penultimatecenter distance of a speed reducer respectively or conversely. 13.Series of reducers according to claim 12, with the characteristic thatthe values of the reduction ratios of the reducers, having two to threereduction steps, are essentially comprised between 8/1 and 100/1. 14.Series of reducers, according to claim 13, with the characteristic thatthe secondary shaft of at least one of the reducers in the series, ishollow.
 15. Series of reducers, according to claim 14, with thecharacteristic that the primary and secondary hollow shafts of one orall reducers are parallel.
 16. Series of reducers, according to claim14, with the characteristic that the primary and secondary hollow shaftsof one or all reducers are perpendicular.
 17. Series of reducers,according to claim 14, with the characteristic that the primary shaft ofone or all stated reducers with hollow shaft, is replaced by flangemotor.
 18. In its capacity as a new industrial product, each gear speedreducer being part of a series of reducers, according to claim 1.